CAIRO, June 3 (Reuters) - Egyptian presidential candidateAhmed Shafiq stepped up his attacks against his MuslimBrotherhood rival in a run-off vote, saying on Sunday theIslamist would drag Egypt into the "dark ages" and threaten therights of women, Christians and others.

The comments from the last Prime Minister of formerpresident Hosni Mubarack, his first public address for more thana week, reflected how divisive the race has become since theformer air force commander made it through last month's firstround into a head-to-head against Mohamed Mursi.

For the bloc of Egyptians who voted for centrist candidatesin last month's first round, the outcome could hardly have beenworse. Many of them worry as much about putting a conservativeIslamist in charge as they do about handing power back to a manwho, like Mubarak, has a military background. Mubarak was jailedfor life on Saturday.

The run-off will be on June 16 and 17. The lead-up to thepoll has been marred by violence, in which several of Shafiq'scampaign offices have been attacked, and many protesters havehit the streets to demonstrate against both candidates.

Sunday's statement appeared mainly to play on the fears ofliberals, Christians (who make up a tenth of Egypt's 82 millionpeople) and women. "Women of Egypt, I will not permit that thepowers of extremism take you back to the dark ages," he said.

Shafiq, 70, who delivered his statement in a five-star hotelon Cairo's outskirts where there was a strong police presence,said the Brotherhood were "liars" and wanted to "penetrate" allinstitutions to create a state in line with their views.

He accused the Brotherhood of making a power grab despitepromising not to run for the presidency, after the organisationalso won more seats in a parliamentary vote than it originallysaid it would seek.

He suggested Mursi would answer to a religious leader notthe people and would create sectarian divisions.

Shafiq had criticised the Brotherhood on May 26 at a newsconference, after initial results showed he was in the run-off.But Sunday's broadside was his fiercest in the campaign so far.It was also timed to coincide with the start of overseas voting.

Mursi, 60, in his news conference on Saturday, had presentedhimself as the "revolutionary" candidate in the race and saidhis rival would rebuild Mubarak's old regime.

CENTRE VOTERS

Mursi has said he would quit the Brotherhood and its partyif elected and insists Christians would enjoy the same rights aseveryone under his presidency, while women would be free to wearwhat they want, with no obligation to put on a Muslim veil.

Mursi has sought to reach out to more centrist candidateswho lost in the first round, namely ex-Brotherhood member AbdelMoneim Abol Fotouh and leftist Hamdeen Sabahy. But he has yet towin explicit backing from either of them.

But those two losing candidates have been vociferousopponents of the "feloul", a derisive Arabic term referring to"remnants" of Mubarak's old order, such as Shafiq. They havevowed to protect the gains achieved since the uprising thattoppled Mubarak on Feb. 11, 2011.

Sabahy and Abol Fotouh, who came third and fourth in thefirst round, secured a total of about 40 percent of votes cast,compared with less than a quarter won by each of Mursi andShafiq. That means the centre-ground voters could be vital forvictory in the run-off.

Turnout was 46 percent of around 50 million eligible voters.

Shafiq has not sought meetings with the other twocandidates, a move that would certainly struggle given theiropposition to "feloul", but his news conference on Saturday, inwhich he took no questions, was clearly aimed at picking offvoters in the centre and those with misgivings about theBrotherhood.

"Choose a president of Egypt who will make a country foreveryone, and not for one sect," he said, adding he was reachingout to every political power "even if it disagrees" with him.

Though he does not apologise for his links to Mubarak, whohe has described as a role model, Shafiq insisted he would notrebuild the old order. He said there could be no return tojailing people for their views and said he would stamp outcorruption, among the issues that drove the anti-Mubarak revolt.

One of his strongest cards is that, as a former militaryman, he is seen by many Egyptians as having the army's backingto help restore order after almost 16 months of turmoil.

"We will work immediately to restore security," he said.

He repeated another of his regular themes, that he hasexperience of office, and also promised poor farmers he wouldensure fair prices for their produce, cancel debts they havewith a state bank and give them health insurance.

To business, he pledged to draw in foreign investors, scaredaway by the turbulence since Mubarak's ouster.